The exact realisation of the Lanczos Method for a quantum Many-Body System

TL;DR

This paper analytically derives the Lanczos process for a quantum spin chain, providing exact formulas for the Lanczos coefficients and insights into ground state properties of many-body systems.

Contribution

It presents an exact analytical solution for the Lanczos method applied to the spin 1/2 XY chain in the thermodynamic limit, including the form of the Lanczos coefficient.

Findings

Lanczos coefficient $eta^2(s)$ has a monotonic variation for positive real s

Finite radius of convergence for the Taylor expansion of the coefficient

Ground state estimates from finite truncations are asymptotic

Abstract

The Lanczos process has been analytically and exactly carried out for the spin 1/2 isotropic XY chain in the thermodynamic limit, yielding a form for the Lanczos coefficient $\beta^2(s)$. This coefficient has a monotonic variation for real positive $s$ and confirms a general theorem on the ground state properties of extensive Many-body Systems. The Taylor expansion of the coefficient about s=0 has a finite radius of convergence, and ground state estimates based on a finite truncation of this are shown to be asymptotic.